How should one properly construct compound nationalities?

In my understanding the word order matters here because what you actually have is adjective + noun. So "Chinese American" is an American (noun) who is adjectivally Chinese. Presumably this means ethnically Chinese, but as you point out in your question, "ethnicity" is not necessarily a relevant interpretation. An American Chinese would be a "Chinese" person who exhibits some quality of American-ness. Americans don't have a typical "ethnicity", but presumably any person of non-Chinese heritage who lived in China could be considered American Chinese, once they attain whatever they need to be considered "Chinese" in the first place.

It comes down to preference. In the USA ethnicity-American is standard. In the UK it is more complicated: British Indian is usually used those of Indian decent resident in the UK or a British citizen, while Anglo-Indian will usually mean a person who has mixed Indian and British ancestry, or someone of British descent who was born or has lived in India.